Normal Random Variable - Random Signals and Processes Primer with MATLAB - page 263

4.8 Numerical Exercises

Exercise 4.1 The normal random variable X has the parameters

X ¼ 5 X 2 ¼ 81 : (4.143)

;

Find the probability that the random variable is in the interval [4, 20] using the erf
and Q functions.

Answer Using ( 4.143 ) the variance of the variable X is:

X 2

s 2 ¼ 81 5 2 ¼ X 2 ¼ 56 : (4.144)

Using ( 4.31 ) and using the MATLAB function erf, we have:

erf

1
2

x 2 m
p s

x 1 m
p s

Pfx 1 X x 2 g¼Pf 4 X 20 g¼ erf

1
2

20 5
2 56

4 5
2 56

¼ erf p erf p

1
2 0

¼ ½ : 9550 ð 0 : 1063 Þ ¼ 0 : 5306 :

(4.145)

The desired probability (Fig. 4.18 ) can also be expressed as:

Pf 4 X 20 g¼PfX> 4 gPfX> 20 g¼ 1 Qðk 1 ÞQðk 2 Þ: (4.146)

From the definition of the Q function ( 4.47 ) and ( 4.146 ), we have:

4 ¼ m k 1 s ; 20 ¼ m þ k 2 s: (4.147)

Fig. 4.18 The probability
and Q function